In his acclaimed book on 20th-century music, The Rest Is Noise, Alex Ross spelt out the future for 21st-century composition as he sees it. "The impulse to pit classical music against pop culture no longer makes intellectual or emotional sense. Young composers have grown up with pop music ringing in their ears," he wrote, predicting that the "possible destination for 21st-century music is a final 'great-fusion': intelligent pop artists and extroverted composers speaking more or less the same language". Or perhaps better still: being more or less the same person.

Certainly that might be the case for Mica Levi, aka Micachu, whose band, Micachu and the Shapes, are Rough Trade's newest arbiters of wildly intelligent pop. The band's forthcoming album Jewellery, produced by Matthew Herbert, is one of the most unusual albums you will hear. It is a fearlessly experimental, effervescent record which blends the organic and the digital, featuring skiffly guitar-strumming, glitchy beats and a vacuum cleaner.

A graduate of Guildhall School of Music and Drama, where she studied composition, Levi was last year commissioned by composer Mark Anthony Turnage to write a piece for the London Philharmonic Orchestra. The result was Interfear, an eight-minute snatch of itchy, avant-garde music inspired, says Levi, by the "bleepy sounds and white noise" that results when trying to tune into a pirate radio station, performed at the Royal Festival Hall in April.

Yet, dressed in a black cut-off T-shirt, grey stonewash drainpipes and converse sneakers in a greasy spoon caff in Brighton, Levi seems markedly different from your average composer.

Barely more than 5ft tall, with an unruly shock of frizzy brown curls and gappy smile, she is tough and streetwise. "Two eggs. Two toast. Bish bash bosh. Done," she remarks after swiftly choosing her breakfast.

She grew up in the Surrey suburb of Guildford. Her mother is a full-time cello teacher, and her father is a lecturer at London's Royal Holloway College, specialising in the subject of Music in the Third Reich: suffice to say music in the Levi household was a serious business.

From a young age she was immersed in music, picking up the violin as a toddler. "I think it's good for your brain to be committed to learning something or other when you are young. It keeps your ears alert and helps you learn self discipline." At sixth form college in Watford she got into the more outré electronic artists such as Aphex Twin, and started making music on computers, staying up all night composing loops on programs such as Logic. After living in halls at Guildhall she moved to Bow, and hooked up with some of the area's abundant MCs.

"We hung out a lot actually," she recalls. "It was quite strange. They were crazy boys, in trouble with the law all the time. We were from such different musical social backgrounds, but it didn't really seem to matter." The result of the immersion was the mix tape Filthy Friends, the sonic equivalent of an invigorating dive into the murk of London past, present and future. Unknown grime MCs such as Ghost Poet and Mayhem compete with Jack Penate, the Streets' Mike Skinner and, bizarrely, Simon Callow reading from Peter Ackroyd's London: The Biography, over a constantly changing selection of electronic beats.

"I just wanted something to dress it up with really," she says of the Ackroyd inclusion, "to give it a bit of context."

The band all studied at Guildhall. Levi and keyboardist Raisa Khan lived together in halls on the Barbican estate where the college is situated, while drummer Mark Pell answered an advert on Levi's MySpace page in late 2007.

Mica was already working with the fastidious composer, producer and polemicist Matthew Herbert, on what was to be an electronic album, before the band was formed. "It felt so fun," she says of the first rehearsals with the trio. "I had never really been in that situation where you could just try out ideas and really work at things." Herbert inspired the band to include samples of anything, from the household vacuum cleaner on Turn Me Well – Levi utilising the device on stage is a highlight of the band's live show – to the ocean.

"A few years ago Matthew made this whole house track out of the sound of Second World War bombs," says Pell. "It is so funny when you see people dancing to it."

Levi's hero is Harry Partch, the 20th-century Californian composer known for his inventive homemade instruments, and she has constructed her own bizarre contraptions, including "the chu", an adapted guitar. "I'm working on another one at the moment using a wooden CD rack with lots of the bits knocked out," she explains. "It has a wheel on it and six strings and finger pads to play bleepy melodies on."

It is this kind of joyful experimentalism that makes the band fit perfectly into Rough Trade's legacy of fostering English eccentrics, from Seventies free-formers This Heat, to the post-structuralist soul of Scritti Politti.

Yet, while Levi embraces her avant-garde tendencies, she does not want to stray too far into the margins with this band. "Pop music has a definite structure; verse, chorus, verse, chorus or whatever," she says. "I think giving yourself limits means you are a lot more creative within your box. You've got to have rules to break."